Junwei Zhao¹, Ruizhu Chen¹, S.P. Rajaguru², Shukur Kholikov^3,4
1 W. W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085
2 Indian Institute of Astrophysics, II Block Koramangala, Bengaluru, 560 034, India
3 National Solar Observatory, 3665 Discovery Dr., Boulder, CO 80303
4 National Research University TIIAME, Kori Niyoziy 39, Tashkent, 100000, Uzbekistan

Helioseismology infers motions beneath the solar surface by measuring the travel times or phases of acoustic waves. The method is sensitive enough to detect the Sun’s slow meridional circulation, but that sensitivity also makes it vulnerable to small observational biases. A direct comparison of 14 years of contemporaneous Doppler observations from the space-based SDO/HMI and the ground-based GONG reveals phase-shift patterns that are alarming. These anomalies can imitate large-scale flows and therefore distort inferences about circulation inside the convection zone, where competing models predict fundamentally different flow structures.

The comparison was made pixel by pixel after the two datasets, each of which lasts 14 years, were processed onto the same solar-disk projection. At each location, the Doppler-velocity time series from HMI and GONG were cross-correlated in the Fourier domain following method given in Ref[1],
,
where represents HMI-observed oscillation signals at location , represents signals observed by GONG. The symbol represents Fourier transform and represents taking conjugate.

Figure 1. Map of relative phase shifts between SDO/HMI and GONG Dopplergrams, obtained between the frequency of 3.0 – 4.0 mHz for 2018 January. Note that the values are saturated near the limbs, but these areas are typically out of our interest in making meridional circulation measurements. The dark box indicates the location of the major phase anomalies, inside which the values are averaged for the temporal evolution shown in Figure 2.

The argument of the averaged complex cross-spectrum gives the relative oscillatory phase shift between the instruments. A uniform timing offset was removed, while the axisymmetric center-to-limb variation^[2] expected from the instruments’ use of different spectral lines was treated separately. What remained should have been nearly uniform across the disk if both systems recorded the same oscillations without spatially dependent artifacts.

Instead, a localized anomaly appears in the northwestern quadrant of the disk. In the representative January 2018 map shown in Figure 1, the strongest patch lies roughly between 0° and 20° longitude and 15° and 35° latitude. In the 3-4 mHz band, its phase amplitude is about 7 milliradians, equivalent to an apparent time shift of roughly 0.3 seconds. Although this sounds small, helioseismic meridional-flow signals are themselves extremely weak. Any acoustic ray path that begins or ends in the affected region can therefore acquire an artificial travel-time contribution comparable to the signal being sought.

Figure 2. (a) Temporal evolution of the phase anomalies measured between the SDO/HMI and GONG Dopplergrams, averaged over the area indicated by the box in Figure 3 for the frequency range of 3.0 – 4.0 mHz for each calendar month. (b) Annual variation of the phase anomalies, averaged from points in (a) using a 1-year window, with error bars representing the standard errors.

The anomaly is visible over most frequency bands between 2 and 6 mHz. Its phase amplitude changes with frequency, but the corresponding time shift remains relatively similar. That behavior points toward a timing discrepancy rather than a frequency-specific solar process. The 3-4 mHz range provides the clearest measurements because solar oscillation power is strongest there, so the long-term analysis concentrates on that band.

The phase anomaly is also not stationary. Figure 2 tracks the monthly and annually averaged phase shift in the anomalous patch from 2010 to 2024. The signal fluctuates substantially, shows some seasonal behavior, and generally increases over time. It does not follow the solar activity cycle in an obvious way. This temporal evolution makes a simple fixed correction inadequate: an empirical correction derived from one interval could misrepresent another.

A second and more widespread problem emerges after the azimuthally averaged center-to-limb component is removed. The 14-year mean map in Figure 3 displays a broad phase gradient that decreases from the northeastern to the southwestern quadrant. Diagonal stripes, probably related to interference patterns in HMI filtergrams, are also visible. The large-scale gradient is especially consequential because helioseismic analysis can interpret a spatial phase trend as motion. It would resemble a southward flow that is stronger west of the central meridian than east of it^[3,4]. It could also create apparent longitudinal flows, making the southern hemisphere seem to rotate slightly faster than the northern hemisphere.

Figure 3. (a) Phase-shift map between SDO/HMI and GONG Dopplergrams averaged between 3.0 – 4.0 mHz throughout the entire analysis period of 14 years, displayed after an azimuthally-averaged CtoL variation is removed. (b) Map of standard errors corresponding to phase-shift map shown in (a), estimated from monthly measurements throughout the 14-year span. (c) Phase shifts averaged in the latitudinal bands of 30°S– 20°S (cyan), 10°S– 10°N (dark), and 20°N–30°N (magenta), displayed as functions of longitude. Shades of different colors indicate the range of errors. (d) Phase shifts averaged in the longitudinal band of 30°E–20°E (cyan), 10°E–10°W (dark), and 20°W–30°W (magenta), displayed as functions of latitude. Note that the map in (a) is in Postel-projected coordinate and the averaging bands to obtain curves in panels c and d are thus not in heliographic coordinates.

These relative phase measurements do not identify a single instrument as more problematic than the other. They only show that HMI and GONG disagree in a structured, time-dependent way, and separate travel-time tests indicate that both datasets contain systematic effects. The discrepancies are stronger on the western side of the solar disk, consistent with the phase gradient and localized patch. Because the phase analysis averages over frequency without separating horizontal wavenumber, it cannot determine precisely which depths are most affected. However, the travel-time inconsistencies extend across a broad range of measurement distances, suggesting that the consequences may reach through much of the convection zone.

The central lesson is that phase-shift anomalies are not merely cosmetic calibration differences. Their location, disk-wide gradient, and temporal variability can all be translated by helioseismic methods into artificial circulation or rotation. Reliable measurements of the Sun’s internal meridional flow will therefore require corrections that are spatially resolved, time dependent, and validated independently for each instrument.

For more details, please refer to our publication Ref. [5].

References

[1] Chen, R., & Zhao, J. 2017, ApJ, 849, 144
[2] Zhao, J., Nagashima, K., Bogart, R.S., Kosovichev, A.G., & Duvall, T.L. Jr., 2012, ApJL, 749, L5
[3] Gizon, L., Cameron, R.H., Pourabdian, M., et al., 2020, Science, 368, 1469
[4] Braun, D.C., Birch, A.C., & Fan, Y., 2021, ApJ, 911, 54
[5] Zhao, J., Chen, R., Rajaguru, S. P., & Kholikov, S. 2026, Solar Phys, 301, 101